§4.3: Random restrictions

In this section we describe the method of applying random restrictions. This is a very “Fourier-friendly” way of simplifying a boolean function.
As motivation, let’s consider the problem of bounding total influence for size-$s$ DNFs. One plan is to use the results from Section 1: size-$s$ DNFs are $.01$-close to width-$O(\log s)$ DNFs, which in turn have total influence $O(\log s)$. This suggests that size-$s$ DNFs themselves have total influence $O(\log s)$. To prove this though we’ll need to reverse the steps of the plan: instead of truncating DNFs to a fixed width and arguing that a random input is unlikely to notice, we’ll first pick a random (partial) input and argue that this is likely to make the width small.

Let’s formalize the notion of a random partial input, or restriction:

Definition 15 For $\delta \in [0,1]$, we say that $\boldsymbol{J}$ is a $\delta$-random subset of $N$ if it is formed by including each element of $N$ independently with probability $\delta$. We define a $\delta$-random restriction on $\{-1,1\}^n$ to be a pair $(\boldsymbol{J} \mid \boldsymbol{z})$, where first $\boldsymbol{J}$ is chosen to be a $\delta$-random subset of $[n]$ and then $\boldsymbol{z} \sim \{-1,1\}^{\overline{\boldsymbol{J}}}$ is chosen uniformly at random. We say that coordinate $i \in [n]$ is free if $i \in \boldsymbol{J}$ and is fixed if $i \notin \boldsymbol{J}$. An equivalent definition is that each coordinate $i$ is (independently) free with probability $\delta$ and fixed to $\pm 1$ with probability $(1-\delta)/2$ each.

Given $f : \{-1,1\}^n \to {\mathbb R}$ and a random restriction $(\boldsymbol{J} \mid \boldsymbol{z})$, we can form the restricted function $f_{\boldsymbol{J} \mid \boldsymbol{z}} : \{-1,1\}^{\boldsymbol{J}} \to {\mathbb R}$ as usual. However it’s inconvenient that the domain of this function depends on the random restriction. Thus when dealing with random restriction we usually invoke the following convention:

(Proving Corollary 18 via Proposition 17 is a bit more elaborate than necessary; see the exercises for a simpler method.)

Corollary 18 lets us bound the total influence of a function $f$ by bounding the (expected) total influence of a random restriction of $f$. This is useful if $f$ is computable by a DNF formula of small size, since a random restriction is very likely to make this DNF have small width. This is a consequence of the following lemma:

Proof: We may assume the initial width of $T$ is at least $w$, as otherwise its restriction under $(\boldsymbol{J} \mid \boldsymbol{z})$ cannot have width at least $w$. Now if any literal appearing in $T$ is fixed to $\mathsf{False}$ by the random restriction, the restricted term $T_{\boldsymbol{J} \mid \boldsymbol{z}}$ will be constantly $\mathsf{False}$ and thus have width $0 < w$. Each literal is fixed to $\mathsf{False}$ with probability $1/4$; hence the probability no literal in $T$ is fixed to $\mathsf{False}$ is at most $(3/4)^w$. $\Box$